Abstract. In this article we prove that for a smooth fiberwise convex Hamiltonian, the asymptotic Hofer distance from the identity gives a strict upper bound to the value at 0 of Mather's ${\beta}$ function, thus providing a negative answer to a question asked by K. Siburg in [K. F. Siburg, Duke Math. J., 92 (2): 295-319, 1998]. However,we show that equality holds if one considers the asymptotic distance defined in [C. Viterbo, Math. Ann., 292 (4): 685-710, 1992].